Tough physics question...

[mfs378]

I wonder if anyone can figure this out.

What is the function that describes an infinitely long rope (lets say it starts at the origin and continues on to positive infinity) whose end is picked up to a finite height at the origin.

I am not sure how to derive this, but am curious because it seems that the function should be infinitely differentible (intuitively). Anyone know how to do this, or know the answer from somewhere?

 

[Anubis]

you can model it with a Fourier sign series or transform. its an infinite equation with an infinite # of variables. ive got a excel spreadsheet that will do it for you round here someplace

 

[mfs378]

That doesn't help.

 

[Anubis]

Originally posted by: mfs378
That doesn't help.

it cant be described with 1 function

 

[mfs378]

Why would you say that?

 

[Stojakapimp]

um, if it's a rope, then it seems like it would go down at 1/x but then it would actually converge with y=0

 

[GaryShandling]

Theres no such thing as "continues on to a posotive infinity". Any question wich uses an un equational method cant be answered by logic or mathmatics because it derives from a philosophical gender.

 

[Anubis]

sorry i thought you were fixing both ends and plucking it at one end like a guitar string

 

[mfs378]

I can see how it would 'look' like 1/x, but you're just pulling that out of the air. And it should reach y=0, because otherwise the entire weight of the string would be supported by the thing which is holding one end up. And since the string is infinitely long, well, you can see why that is unreasonable.

 

[mfs378]

Originally posted by: GaryShandling
Theres no such thing as "continues on to a posotive infinity". Any question wich uses an un equational method cant be answered by logic or mathmatics because it derives from a philosophical gender.

Umm... what? Infinity is in plenty of physically relevant mathematically describable systems.

 

[Stojakapimp]

well yes I just pulled 1/x out of the air, but the string will be a variation of that. You have to take string tension, weight, height, the density, and all of that into account. So maybe it'll look something like 3/52x. but it will still have the same curvatures are 1/x

 

[RaynorWolfcastle]

hmmm... well you'll have to remember that there are three forces being applied on each infinitely small portion of rope (call it dx if you want)

there will be a force exerted by string on the left side, a force by the string on the right side and gravity pulling it down. I could probably figure it out with some thought, but I bet you'll end up with a dirty integral to solve

 

[mfs378]

Dirty integrals are no problem...

And it won't be anything like 1/x or 3/52x (at least if my thinking is correct) because in that case, the string wouldn't be touching the ground anywhere, which is impossible, since the string is not perfectly rigid.

 

[Stojakapimp]

well then in that case, you are going to have to find a function where y<0 for small values of x, and after a while, every value of x will give you y=0.

 

[GaryShandling]

Originally posted by: mfs378

Originally posted by: GaryShandling
Theres no such thing as "continues on to a posotive infinity". Any question wich uses an un equational method cant be answered by logic or mathmatics because it derives from a philosophical gender.

Umm... what? Infinity is in plenty of physically relevant mathematically describable systems.

Infinity is what man describes as "unknown" you cant create logic and equation from non factual substance. Any mathmatical sum which derives from an infinity remark or which contains it has no SUM.

 

[notfred]

Originally posted by: mfs378
because otherwise the entire weight of the string would be supported by the thing which is holding one end up. And since the string is infinitely long, well, you can see why that is unreasonable.

You're saying that it would be unreasonable for something to support the weight of an infinitely long string. However, you have no problems with the existance of an infinitly long string in the first place. Am I correct here?

 

[Spencer278]

You need more info and a clear question. Should we assume their is gravity and y = 0 is the ground. In my physics class ropes were always assumed to have no mass and be infinitly flexible so using those two assumetions your function would be y= originail hieght. Assuming it is not the trivial case then you need to equations to model your rope I would guess for the first part when the rope is in the air, f, it would be a f = 1/x type and on the ground, g, it would be y = ( radius of the rope ). Then just make the functions piecewise continuos be stating f = g and that is where the rope touches the ground.

 

[Armitage]

look up catenary ... that's the shape described by a rope (cable, chain, etc.) hanging from two supports of equal heights. The derivation of the system you describe should be similar.

 

[DrPizza]

It's been a while since I've worked on such problems, but I believe hyperbolic functions would describe the shape.

 

[RossGr]

Originally posted by: ergeorge
look up catenary ... that's the shape described by a rope (cable, chain, etc.) hanging from two supports of equal heights. The derivation of the system you describe should be similar.

I'll second this.

 

[RossGr]

Originally posted by: GaryShandling

Originally posted by: mfs378

Originally posted by: GaryShandling
Theres no such thing as "continues on to a posotive infinity". Any question wich uses an un equational method cant be answered by logic or mathmatics because it derives from a philosophical gender.

Umm... what? Infinity is in plenty of physically relevant mathematically describable systems.

Infinity is what man describes as "unknown" you cant create logic and equation from non factual substance. Any mathmatical sum which derives from an infinity remark or which contains it has no SUM.

You really should just stay out of math disscussions, clearly you do not think it has any meaning. Why even comment?

 

[GaryShandling]

Originally posted by: RossGr

Originally posted by: GaryShandling

Originally posted by: mfs378

Originally posted by: GaryShandling
Theres no such thing as "continues on to a posotive infinity". Any question wich uses an un equational method cant be answered by logic or mathmatics because it derives from a philosophical gender.

Umm... what? Infinity is in plenty of physically relevant mathematically describable systems.

Infinity is what man describes as "unknown" you cant create logic and equation from non factual substance. Any mathmatical sum which derives from an infinity remark or which contains it has no SUM.

You really should just stay out of math disscussions, clearly you do not think it has any meaning. Why even comment?

You wont understand, dont worry about it.

 

[dullard]

This is a very difficult problem to do correctly - since it will depend on the properties of the rope. For example, an infinitely thin rope held up by an infinitely thin holder at x=0 will have a kink at x=0. Thus it will be differentiable everywhere except at x=0. However that isn't a realistic senario. All ropes have non-zero thicknesses, and will be held by non-zero width holders. Thus there will be a turning radius that the rope cannot exceed. Thus it will have a smooth curve at x=0 and will be differentiable everywhere. The difficulty comes since this turning radius will be a function of the rope material properties as well as the diameter of the rope (thus the shape function will depend on these as well).

Too many people here are thinking the rope will be off the ground everywhere - which won't be the case. Go pick up a long rope - does the other end lift off the ground? No. Thus you really don't even need to consider an infinitely long rope - just long enough so that the end is flat on the floor when the other end is lifted.

We did similar problems in a Mechanics of Materials class. But that was 7 years ago and I haven't used it since, so I don't have anything memorized to help you.

 

[WarCon]

dullard is perfectly correct, if there is no tensioning force the rope will have a drop off back to "level" based on its properties and the properties you assign to the environment. Equation gets pretty complex.

 

[glugglug]

The cable will form half of a catenary curve (google it), until it eventually touches the ground.

How far away it stays on the ground will depend on how hard/high you are pulling the end.